Question

A hand fan is made up of cloth fixed in between the metallic wires. It is in
the shape of a sector of a circle of radius 21 cm and of angle 120° as
shown in the figure. Calculate the area of the cloth used and also find the
total length of the metallic wire required to make such a fan.​

Answer 1

Given:

A hand fan which is in the shape of a sector of a circle of radius 21 cm and of angle 120°, is made up of cloth fixed in between the metallic wires

To find:

(i) Area of the cloth used

(ii) The total length of the metallic wire required to make such a fan

Solution:

(i) Finding the area of the cloth used:

The radius of the sector of the circle, r = 21 cm

The measure of the central angle, θ = 120°

Here to get the area of the cloth used, we have to find the area of the sector.

So, we know the formula of the area of a sector when θ is given in terms of degrees is given as,

$\boxed{\boxed{\bold{Area \:of \:a \:sector = \frac{\theta }{360}\times \pi r^2 }}}$

By substituting the given values in the formula, we get

$Area \:of \:a \:sector = \frac{120 }{360}\times \frac{22}{7} \times 21^2 }}} = \frac{1 }{3}\times \frac{22}{7} \times 441 }}} = \frac{9702}{21} = 462\:cm^2$

Thus, the area of the cloth used is 462 cm².

(ii) Finding the total length of the metallic wire:

We will first find the arc length of the hand-made fan i.e.,

$Arc \:length = \frac{\theta}{360} \times 2\pi r = \frac{120}{360} \times 2\times \frac{22}{7} \times 21 = \frac{1}{3} \times 2\times \frac{22}{7} \times 21 = 2 \times 22 = 44 \: cm$

Now,

The required length of the metallic wire is given by,

= sum of all sides of the hand-made fan

= [2 × radius] + Arc length

substituting the value of radius and arc length

= [2 × 21] + 44

= 42 + 44

= 86 cm

Thus, the total length of the metallic wire required to make such a fan is 86 cm.

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